Let $X$ be the space of all sequences of complex numbers $(x_1,x_2,\ldots)$ such that $\lim_{n}{x_{n}}$ exists, and consider in this space the norm $\|x\|=\sup_{n}{|x_n|}$.
I need to find a linear continuous functional in $X$, that doesn't attain its norm in the closed unit ball.
I thought a functional like, $f:X\to\mathbb{K}$, defined as $$f(x)=\sum_{n\geq1}{\frac{x_n}{n!}}.$$
Ok, in your case norm is easily attainable if you take $x = (1,1,1,\ldots)$.
Proper answer: take $$f(x) = \sum_{n\ge 1} \frac{e^{i\cdot n}}{n!}\cdot x_n,$$ this way to get $x$ which maximizes $|f(x)|$ you need to choose $x_i = e^{- i\cdot n}$, but a sequence with such elements doesn't converge.